Let \((\mathcal {B},|\cdot |)\)  be a separable Banach space with a Borel \(\sigma \) -algebra \({B}\) .
We denote by \(B(\mathcal {B})\)  the space of all bounded linear operators on \(\mathcal {B}\)  and also use \(|\cdot |\)  to refer to the operator norm on \(B(\mathcal {B})\) .
The strong operator topology \(SOT(\mathcal {B})\)  on \(B(\mathcal {B})\)  is the topology generated by the sub-base
\(\lbrace V_{x,y,\epsilon }=\lbrace T:|T(x)-y|<\epsilon \rbrace :x,y\in \mathcal {B}, \epsilon >0\rbrace \) .
A sequence of bounded linear operators \(T_n:n>0\)  converges to \(T\in B(\mathcal {B})\)  on \(SOT(\mathcal {B})\)  if and only if \(T_n(x)\rightarrow T(x)\)  as \(n\rightarrow \infty \)  for every \(x\in \mathcal {B}\) .
The strong \(\sigma \) -algebra on \(B(\mathcal {B})\)  is the Borel \(\sigma \) -algebra of the strong operator topology \(SOT(\mathcal {B})\) .
For a function \(L:\Omega \rightarrow B(\mathcal {B})\) , we call \(L\)  is strongly measurable, if \(\omega \mapsto L(\omega )v\)  is measurable for every \(v\in \mathcal {B}\) .
We refer that \(L\)  is strongly measurable if and only if it is measurable with respect to the strong \(\sigma \) -algebra on \(B(\mathcal {B})\)  (see [1]}).
